Chapter 1 Fundamentals - Seoul National University · 2018. 1. 30. · Chapter 1 Fundamentals...

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Chapter 1

Fundamentals

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Chapter 1 Fundamentals

Contents1.1 Scope of Fluid Mechanics

1.2 Historical Perspective

1.3 Physical Characteristics of the Fluid State

1.4 Units and Density

1.5 Compressibility, Elasticity

1.6 Viscosity

1.7 Surface Tension, Capillarity

1.8 Vapor Pressure

Objectives

- Present concept of fluidity

- Introduce fundamental properties of fluid

3/92

1.1 Scope of Fluid Mechanics

• Problems of water supply

flood prevention

navigation need to know fluid phenomena

water power

irrigation

4/92


• d'Alembert (1744)

"The theory of fluids must necessarily be based upon experiment"

• d'Alembert paradox theory - ideal, inviscid fluid

practice - real fluid (viscous)

• Two schools theoretical group → hydrodynamics

practical group → hydraulics

• Navier and Stokes

→ general equations for viscous fluid → equation of motion

5/92


[Re] Navier-Stokes equation

Claude-Louis Navier (1785-1836, French engineer) and George Gabriel

Stokes (1819-1903, UK mathematician & physicist)

- one continuity equation + three momentum equations

- model the weather, ocean currents, water flow in a pipe, the air's flow

around a wing, and motion of stars inside a galaxy

- design of aircraft and cars, the study of blood flow, the design of power

stations, the analysis of pollution,

- exact solution - one of the seven most important open problems in

mathematics

6/92


• New problems in modern times

- Dispersion of man's wastes in lakes, rivers, and oceans

→ Environmental Fluid Mechanics (Hydraulics)

• state: solid

liquid increasing spacing increasing inter

fluid gaseous and latitude of molecular cohesive

plasma particle motion force

• fluid – continuum → no voids or holes

7/92


8/92


stressstrain

solid fluid

tension

elastic deformation

→ permanent distortion

unable to support tension

(surface tension)

compressionelastic deformation

(compressible fluid)

shear (tangential forces)

permanent distortion or flo

w (change shape) to infini

tesimal shear stress

9/92


dvdy

τ µ=

10/92


stress

real fluid (viscous fluid)

ideal fluid

(non-viscous

fluid)

in motion at restat rest and

in motion

compression

(pressure)○ ○ ○

shear ○ × ×

11/92


incompressible fluid compressible fluid

① Compressibility is of small important. ① Compressibility is predominant.

② Liquids and gases may be treated

similarly.

② Behavior of liquids and gases is quite

dissimilar.

③ Fluid problems may be solved with

the principles of mechanics.

③ Thermodynamics and heat transfer

concepts must be used as well as

principles of mechanics.

12/92


• Properties of pressure (compression)

① Pressure must be transmitted to solid boundaries normal to those

boundaries.

② At a point, pressure has the same magnitude in all directions.

→ Pressure is a scalar quantity.

* Fluid does not resist any small shearing stress → "Flow occurs"

13/92


Weight.W Volγ=

14/92


0F∑ =

(a)1 3 sin 0xF p dz p ds θ∑ = − =

2 3/ 2 cos 0zF p dx gdxdz p dsρ θ∑ = − − =

[Pf]

Apply Newton's law for static equilibrium

(b)

sindz ds θ=cosdx ds θ=

1 3 1 3( ) : sin sin 0a p ds p ds p pθ θ∴ − = → =

Substitute following relations into Eq. (a) & (b)

15/92


2 3( ) : cos cos cos 02dzb p ds g ds p dsθ ρ θ θ− − =

2 312

p p gdzρ∴ = +

0dz → 2 3p p≈

1 2 3p p p∴ = = ( 0)dx dz= =

As then

at a point

16/92


• SI units - SI system – metric system

f- Frequency ( ): hertz (HZ = s-1)

F

F ma=

2/ /a v t L t= = 2Lt− /v L t= 1Lt−

21kg m/s 1N( )F Newton∴ → ⋅ =

• Force,

→ introduce Newton's 2nd law of motion

Force = mass × acceleration

(m/s2)

(m/s)

17/92


Dimension SI unit English system (FSS)

Length (L) metre (m) feet (ft)

Mass (M) kilogram (kg) slug (-)

Time (t) second (s) second (s)

Temp. (T) kelvin (K) degree Rankine (°R)

18/92


E2 2kg m /s ( )E FL J Joule= → ⋅ =

P2 3/ / kg m sP E t J s= → = ⋅

p ,σ τ2 2/ N/m Pa (pascal) kg/m sp F A= → = = ⋅

T

• Energy, (work)

• Power,

• Pressure, ; Stress,

• Temperature, : degree Celsius (°C)

19/92


ρ

3kg/mMV

ρ = →

• Density,

= mass per unit volume

~ depends on the number of molecules per unit of volume

~ decreases with increasing temperature

γ

3 2 2N/m kg/m sWV

γ = → = ⋅

• Specific weight (weight density),

= weight (force) per unit volume

20/92


(1.1)

W Mg=g

gγ ρ∴ =

[Re](Newton’s 2nd law of motion)

= acceleration due to gravity

1/ ρ• Specific volume=volume per unit mass=

• Specific gravity, s.g. , ~ r. d. (relative density)

= ratio of density of a substance to the density of water at a specified

temperature and pressure

. . f fw w

s gρ γρ γ

= =

21/92


[Re] s.g. of sea water = 1.03

s.g. of soil = 2.65

s.g. of mercury = 13.6

( )F ( )M• Advantage of SI system and English FSS system

① It distinguishes between force and mass .

② It has no ambiguous definitions.

22/92


SI English system

1,000 kg/m3 1.94 slugs/ft3

9,806 N/m3 62.4 lb/ft3

9.81 m/s2 32.2 ft/s2

ρ

γ

g

23/92


α

β,γ Γ

,δ ∆εζη

,θ Θ

ι

• Greek Alphabet

Alpha angle

Beta [beitə] angle

Gamma specific weight, circulation

Delta thickness of boundary layer

Epsilon eddy viscosity, height of surface roughness

Zeta

Eta

Theta

Iota [aioutə]

Kappa [kæpə]κ

24/92


,λ Λ

µ

νξοπ

ρ

,σ ∑

τ

Lambda

Mu [mju:] dynamic viscosity

Nu kinematic viscosity

Xi [gzai, ksai] vorticity

Omicron

Pi [pai]

Rho mass density

Sigma Sigma Xi, Scientific Research Society, 1886

honor society for scientists & engineers

Tau shear

Upsilon,υ ϒ

25/92


,ϕ Φ

χ

,ψ Ψ

,ω Ω

Phi [fai] Phi Beta Kappa

Chi [kai]

Psi [psai, sai] stream function

Omega angular velocity

• Prefixes

E exa 1018

P peta 1015

T tera 1012

G giga 109

M mega 106

26/92


µ

M mega 106

k kilo 103

h hecto 102

da deca 101

d deci 10-1

c centi 10-2

m milli 10-3

micro 10-6

n nano 10-9

p pico 10-12

f femto 10-15

a atto 10-18

27/92


• Elastic behavior to compression

• Compressibility ≡ change in volume due to change in pressure

solid - modulus of elasticity, E (N/m2)

fluid - bulk modulus

28/92


29/92


1

dVdpV

∝1

dVdp EV

→ = −

1

1

( , )dp dpE V const fn p TdV dVV

= − = − ≠ = → p E↑ → ↑

1

1 1dVCE V dp

= = −

• Stress-strain curve (E↑, difficult to compress)

= modulus of compressibility (m2/N)

Minus means that increase in

pressure causes decrease in

volume

30/92


, 1E C= ∞ ρ

[Re] large E/small C → less compressible

• incompressible fluid (inelastic):

→ constant density =const.

~ water

• compressible fluid

→ changes in density → variable density

~ gas

31/92


Pressure

106 N/m2

Temperature, °C

0° 20° 50° 100° 150°

0.1 1950 2130 2210 2050

10.0 2000 2200 2280 2130 1650

30.0 2110 2320 2410 2250 1800

100.0 2530 2730 2840 2700 2330

32/92


E

E

- increases as pressure increases.

- is maximum at about 50 °C.

→ The water has minimum compressibility at about 50 °C.

1dpE VdV

= −

1

V pV E∆ ∆

≈ −

2 1 2 1

1

V V p pV E− −

≈ −

• For the case of a fixed mass of liquid at constant temperature

33/92


Compressibility Modulus of Elasticity, E (kPa)

steel 1/80 of water water 2,170,500

mercury 1/12.5 of water sea water 2,300,000

nitric acid 6 of water mercury 26,201,000

34/92


62,200 10 PaE = ×6

2 1 7 10 Pap p= + ×

2 1 2 1

1

0.0032V V p pV E− −

∴ ≈ − = −

2 1(1 0.0032)V V∴ = −

0.3%V∆ ≈

[Ex] For water; @ 20˚C

decrease

→ water is incompressible

63

7 10 Pa 70101.3 10 Pap×∆ = ≈

×기압

35/92

1.6 Viscosity

[Re] From Wikipedia

Viscosity is a measure of the resistance of a fluid which is being

deformed by either shear stress or tensile stress.

Viscosity ~ "thickness" or "internal friction"

▪ water ~ "thin", having a lower viscosity

▪ honey ~ "thick", having a higher viscosity

The less viscous the fluid is, the greater its ease of movement (fluidity).

Viscosity describes a fluid's internal resistance to flow and may be

thought of as a measure of fluid friction.

dvdy

τ µ=

36/92

1.6 Viscosity

For example, high-viscosity felsic magma will create a tall, steep

stratovolcano, because it cannot flow far before it cools, while low-

viscosity mafic lava will create a wide, shallow-sloped shield volcano.

All real fluids (except superfluids) have some resistance to stress and

therefore are viscous, but a fluid which has no resistance to shear stress

is known as an ideal fluid or inviscid fluid.

[Re] super fluid – a fluid having frictionless flow, and other unusual

properties

37/92

1.6 Viscosity

• Two types of fluid motion (real fluid)

1) laminar flow:

- viscosity plays a dominant role

- fluid elements or particles slide over each other in layers (laminar)

- molecular diffusion

[Ex] flow in a very small tube, a very thin flow over the pavement, flow in

the laminar flow table

38/92

1.6 Viscosity

Re Vdν

=

where V = flow velocity; d = characteristic length; n = kinematic viscosity

Diameter of pipe,Depth of stream

2) turbulent flow:

- random or chaotic motion, eddies of various sizes are seen

- common in nature (streams, rivers, pipes)

- large scale mixing between the layers

• Reynolds number

[Ex] flows in the water supply pipe, flows in the storm sewer pipe, flows in

the and canals and streams

39/92

1.6 Viscosity

• Reynolds experiments

laminar flow: Re < 2,100

transition: 2,100 4,000

The same fluid withdifferent velocity

40/92

1.6 Viscosity

Spiralsecondaryflow

Secondaryflow

41/92

1.6 Viscosity

42/92

1.6 Viscosity

cubestreamline

source

sink

43/92

1.6 Viscosity

Smoothsurface

Eddyingmotion

44/92

1.6 Viscosity

45/92

1.6 Viscosity

Symmetric shape,No separation

46/92

1.6 Viscosity

47/92

1.6 Viscosity

Separation, eddyformation

Growthof eddy

48/92

1.6 Viscosity

49/92

1.6 Viscosity

50/92

1.6 Viscosity

• laminar flow

2 1d d dvdt dv dtdy dy dy−

= = = yxdGdyζτ =

2 2 1 1

2 1 2 1

;( )

d v dt d v dtd d v v dt

= =− = −

• strain = relative displacement [Cf] solid mechanics

[Re] total angular displacement

51/92

1.6 Viscosity

no velocity at theboundary (no slip)

52/92

1.6 Viscosity

τ• Experiment has shown that, in many fluids, shearing (frictional) stress

per unit of contact area, is proportional to the time rate of relative

strain.

/dv dvdt dtdy dy

τ∴ ∝ =

dvdy

τ µ= →

(velocity gradient)

Newton’s equation of viscosity (1.2)

µwhere = coefficient of viscosity

= dynamic (absolute) viscosity

Large → sticky, difficult to flowµ

53/92

1.6 Viscosity

• viscosity = measure of fluid's resistance to shear or angular

deformation

= internal resistance of a fluid to motion (fluidity)

[Re] Friction forces result from

- cohesion for liquid

- momentum interchange between molecules for gas

[Re] angular deformation due to tangential stress

54/92

1.6 Viscosity

dy

x

55/92

1.6 Viscosity

du duu y t u t y tdy dy

= + ∆ ∆ − ∆ = ∆ ∆

/du duy t y tdy dy

= ∆ ∆ ∆ = ∆

• rate of angular deformation

(i) displacement of AB relative to CD

(ii) angular displacement of AC

(iii) time rate of angular deformation

/du dut tdy dy

= ∆ ∆ =

dvdy

τ µ=

56/92

1.6 Viscosity

µ• dynamic viscosity,

/F Aτ =

[ ] 2 2 1 2 2/ kg/(m s ) PaMLT L ML Tτ − − − = = ⋅ = 1

1dv LT Tdy L

−− = =

[ ]1 2

1 1 21/ kg/m s N s / m Pa s

dv ML T ML Tdy T

µ τ− −

− −−

∴ = = = = ⋅ = ⋅ = ⋅

11 ( ) 10 Pa spoises Poiseuille −⇒ = ⋅

57/92

1.6 Viscosity

ν• kinematic viscosity,µνρ

=

[ ]1 1

2 1 23 m /s

ML T L TML

ν− −

−−

= = =

(1.3)

1 m2/s = 104 stokes =106 centistokes

,τ µ

τ τ

• Remarks on Eq. (1.2)

① are independent of pressure. [Cf] friction between two moving

solids

② Shear stress (even smallest ) will cause flow (velocity gradient).

58/92

1.6 Viscosity

0 0dvdy

τ= → = µregardless of

③ Shearing stress in viscous fluids at rest will be zero.

dvdy

≠ ∞ τ→ ≠ ∞④ At solid boundary, ( (no infinite shear))→ Infinite shearing stress between fluid and solid is not possible.

⑤ Eq. 1.2 is limited to laminar (non-turbulent) fluid motion in which

viscous action is predominant.

[Cf] turbulent flow

εε µ

dvdy

τ ε=

where = eddy viscosity

59/92

1.6 Viscosity

⑥ Velocity at a solid boundary is zero.

→ No slip condition (continuum assumption)

• Newtonian and non-Newtonian fluids

i) Newtonian fluid ~ water

ii) Non-Newtonian fluid ~ plastic, blood, suspensions, paints, polymer

solutions → rheology

60/92

1.6 Viscosity

61/92

1.6 Viscosity

1dvdy

τ τ µ− =

ndvKdy

τ

=

1τ

1n >

1n <

• Non-Newtonian fluid

1) plastic, = threshold

2) Shear-thickening fluid

Shear-thinning fluid

• Couette flow: laminar flow in which the shear stress is constant

thin fluid film between two large flat plates

thin fluid film between the surfaces of coaxial cylindersdv Vdy h

=

Vh

τ µ∴ =

~ linear velocity gradient

~ constant

62/92

1.6 Viscosity

63/92

1.6 Viscosity

( ) dvdy

τ µ ε= +

ε

• Turbulent flow

= eddy viscosity = viscosity due to turbulent factor

64/92

1.6 Viscosity

gas liquid

main cause of

viscosity

exchange of molecule's momentum →

interchange of molecules between

the fluid layers of different velocities

intermolecular cohesion

effect of

temperature

variation

temp↑ → molecular activity↑

→ viscosity↑ → shearing stress↑

temp↑ → cohesion↓

→ viscosity↓ → shear stress↓

65/92

1.6 Viscosity

Two layers tendto stick togetheras if there is some viscosity between two.

[Re] Exchange of momentum

fast-speed layer (FSL)

molecules from FSL speed up molecules in LSL

molecules from LSL slow down molecules in FSL

low-speed layer (LSL)

66/92

1.6 Viscosity

mv

1) exchange of momentum : exchange momentum in either direction from

high to low or from low to high momentum due to random motion of

molecules

2) transport of momentum : transport of momentum from layers of high

mome

(high velocity, ) to layers of low momentum

67/92


nf

σ

• surface tension

- occur when the liquid surfaces are in contact with another fluid (air) or

solid

- (relative sizes of intermolecular cohesive and adhesive forces to

another body)

- as temp↑ → cohesion↓ → ↓ ☞ Table A2.4b, p. 694

• some important engineering problems related to surface tension

- capillary rise of liquids in narrow spaces

- mechanics of bubble formation

- formation of liquid drops

- small models of larger prototype → dam, river model

68/92


σ /F L• surface tension, ( , N/m)- force per unit length

- force attracting molecules away from liquid

0F∑ = , , ,a b c d

0( ) 2 sin 2 sinip p dxdy dy dxσ α σ β− = +

Consider static equilibrium

(forces normal to the element )

where pi = pressure inside the curvature; po = pressure inside the

curvature1

sin ,2dxR

α =2

sin2dyR

β = [ ]12( sin )dx R α=

01 2

1 1ip p R R

σ

∴ − = +

(1.4)

69/92


70/92


• Cylindrical capillary tube

- due to both cohesion and adhesion

cohesion < adhesion → rise (water)

cohesion > adhesion → depression (mercury)

71/92


72/92


1 2R R R= = ≈

0p hγ= −

0ip =

For a small tube, given conditions are as follows

(liquid surface section of sphere) ← Ch. 2

(hydrostatic pressure)

(atmospheric)

01 2

1 1ip p R R

σ

− = + 2h

Rγ σ∴ =

Substitute above conditions into Eq. 1.15: (1.4)

cosr R θ=2 2 cos

/ cosh

r rσ θγ σ

θ∴ = =

2 coshr

σ θγ

=

By the way,

(1.5)

73/92


h r hθr ≤

in which = capillary rise → ↑ → ↓

= angle of contact

= radius of tube 2.5 mm for spherical form

12r > h[Ex] water and mercury → Fig. 1.11

If mm, is negligible for water.

74/92


cohesion > adhesion

→ depression

75/92


• Pressure measurement using tubes in hydraulic experiments

→ Ch.2 manometer

~ capillarity problems can be avoided entirely by providing tubes large

enough to render the capillarity correction negligible.

• Fomation of curved surface, droplet

- At free liquid surface contacting the air, cohesive forces at the outer

layer are not balanced by a layer above.

→The surface molecules are pulled tightly to the lower layer.

→Free surface is curved.

[Ex] Surface tension force supports small loads (water strider).

76/92


77/92


0 1.0ip p− =

0.0728σ =At 20°C, N/m ← App. 2

[IP 1.10] For a droplet of water (20 °C), find diameter of droplet

Given: kPa

01 2

1 1 2ip p R R R

σσ

− = + =

∴1R⋅

0.000146m 0.146mmR∴ = = 0.292d→ =

[Sol]

(1.4)

1×103 N/m2 = 2(0.0728)

mm

78/92


[IP 1.11] Find height of capillary rise in a clean glass tube of 1 mm

diameter if the water temperature is 10°C or 90°C.

[Sol]

From App. 2 Table A 2.4b;

σ γ

σ γ@ 10°C =0.0742 N/m, = 9.804 kN/m3

@ 90°C =0.0608 N/m, = 9.466 kN/m3

79/92


Use Eq. 1.16

For water, 0θ =

2 coshr

σ θγ

=

1̀02(0.0742)(1) 0.030m=30mm

9804(0.0005)h∴ = =

902(0.0608)(1) 0.026m=26mm

9466(0.0005)h = =

(1.5)

80/92

1.8 Vapor Pressure

• vapor pressure = partial pressure exerted by ejected molecules of liquid

→ Table A2.1 and A2.4b

• liquids ~ tend to vaporize or evaporate due to molecular thermal

vibrations (molecular activity)

→ change from liquid to gaseous phase

temperature↑ → molecular activity↑ → vaporization↑ → vapor pressure↑

81/92

1.8 Vapor Pressure

55.2 kPavp =2.34 kPavp =

0.00017 kPavp =

• volatile liquids:

~ easy to vaporize → high vapor pressure

gasoline: at 20 °C

water: at 20 °C

mercury: at 15.6 °C

• mercury : low vapor pressure and high density = difficult to vaporize

→ suitable for pressure-measuring devices

82/92

1.8 Vapor Pressure

In the interior and/or boundariesof a liquid system

High velocity region

• Cavitation: App. 7 (p. 672)

In a flow fluid wherever the local pressure falls to the vapor pressure of

the liquid, local vaporization occurs.

→ Cavities are formed in the low pressure regions.

→ The cavity contains a swirling mass of droplets and vapor.

→ Cavities are swept downstream into a region of high pressure.

→ Then, cavities are collapses suddenly.

83/92

1.8 Vapor Pressure

→ surrounding liquid rush into the void together

→ it causes erosion (pitting) of solid boundary surfaces in machines, and

vibration

→ boundary wall receives a blow as from a tiny hammer

84/92

1.8 Vapor Pressure

85/92

1.8 Vapor Pressure

• Prevention of cavitation

~ cavitation is of great importance in the design of high-speed hydraulic

machinery such as

turbines, pumps, in the overflow and underflow structures of high dams,

and in high-

speed motion of underwater bodies (submarines, hydrofoils).

→ design improved forms of boundary surfaces

→ predict and control the exact nature of cavitation → set limits

Bodycavitation

86/92

1.8 Vapor Pressure

87/92

1.8 Vapor Pressure

atm vp p≤

• Boiling:

= rapid rate of vaporization caused by an increase in temperature

= formation of vapor bubbles throughout the fluid mass

~ occur (whatever the temperature) when the external absolute pressure

imposed on the

liquid is equal to or less than the vapor pressure of the liquid

~ boiling point = f (imposed pressure, temp.)

→ boiling occurs

88/92

1.8 Vapor Pressure

altitude

(El. m)

Temp.

(°C)

(kPa),

absolute

(kPa),

absolute

boiling point

(°C)remark

m.s.l. 100 101.3 101.3 100

12,000 60 19.9 19.4 60 undercooked

vp atmp

Table A2.4b

Table A2.5b

89/92

1.8 Vapor Pressure

vp• Evaporation: When the space surrounding the liquid is too large, the

liquid continues to

vaporize until the liquid is gone and only vapor remains at a pressure

less than or equal.

[IP 1.12] For a vertical cylinder of diameter 300 mm, find min. force that

will cause the water boil.

90/92

1.8 Vapor Pressure

Patm=100 kpa

Water @ 70˚C

91/92

1.8 Vapor Pressure

vp

'vp p≤

' 100 31.16FpA

∴ = − =

2(0.3)(100 31.16) 4.87 kN4

F π∴ = − =

[Sol] From Table A2.4b; =31.16 kPa at 70 °C

=31.16, For water to boil;

92/92

1.8 Vapor Pressure

Homework Assignment # 1

Due: 1 week from today

Prob. 1.2

Prob. 1.10

Prob. 1.27

Prob. 1.46

Prob. 1.49

Prob. 1.58

Prob. 1.69

Prob. 1.72

Prob. 1.82

Chapter 1Chapter 1 Fundamentals1.1 Scope of Fluid Mechanics��1.2 Historical Perspective��1.2 Historical Perspective��1.2 Historical Perspective��1.3 Physical Characteristics of the Fluid State��1.3 Physical Characteristics of the Fluid State��1.3 Physical Characteristics of the Fluid State��1.3 Physical Characteristics of the Fluid State��1.3 Physical Characteristics of the Fluid State��1.3 Physical Characteristics of the Fluid State��1.3 Physical Characteristics of the Fluid State��1.3 Physical Characteristics of the Fluid State��1.3 Physical Characteristics of the Fluid State��1.4 Units and Density��1.4 Units and Density��1.4 Units and Density��1.4 Units and Density��1.4 Units and Density��1.4 Units and Density��1.4 Units and Density��1.4 Units and Density��1.4 Units and Density��1.4 Units and Density��1.4 Units and Density��1.5 Compressibility, Elasticity��1.5 Compressibility, Elasticity��1.5 Compressibility, Elasticity��1.5 Compressibility, Elasticity��1.5 Compressibility, Elasticity��1.5 Compressibility, Elasticity��1.5 Compressibility, Elasticity��1.5 Compressibility, Elasticity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.6 Viscosity��1.7 Surface Tension, Capillarity��1.7 Surface Tension, Capillarity��1.7 Surface Tension, Capillarity��1.7 Surface Tension, Capillarity��1.7 Surface Tension, Capillarity��1.7 Surface Tension, Capillarity��1.7 Surface Tension, Capillarity��1.7 Surface Tension, Capillarity��1.7 Surface Tension, Capillarity��1.7 Surface Tension, Capillarity��1.7 Surface Tension, Capillarity��1.7 Surface Tension, Capillarity��1.7 Surface Tension, Capillarity��1.8 Vapor Pressure��1.8 Vapor Pressure��1.8 Vapor Pressure��1.8 Vapor Pressure��1.8 Vapor Pressure��1.8 Vapor Pressure��1.8 Vapor Pressure��1.8 Vapor Pressure��1.8 Vapor Pressure��1.8 Vapor Pressure��1.8 Vapor Pressure��1.8 Vapor Pressure��1.8 Vapor Pressure��

Chapter 1 Fundamentals - Seoul National University · 2018. 1. 30. · Chapter 1 Fundamentals...

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